Periodic projective resolutions and Hochschild cohomology for basic hereditary orders

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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Hochschild Cohomology of Minimal Hereditary Orders

Journal of Algebra ◽

10.1006/jabr.1995.1273 ◽

1995 ◽

Vol 176 (3) ◽

pp. 786-805 ◽

Cited By ~ 2

Author(s):

K. Sanada

Keyword(s):

Hochschild Cohomology ◽

Hereditary Orders

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Comparison morphisms between two projective resolutions of monomial algebras

Revista de la Unión Matemática Argentina ◽

10.33044/revuma.v59n1a01 ◽

2017 ◽

pp. 1-31

Author(s):

María Julia Redondo ◽

Lucrecia Román

Keyword(s):

Lie Algebra ◽

Hochschild Cohomology ◽

Cohomology Groups ◽

Efficient Tool ◽

Simple Description ◽

Monomial Algebra ◽

Cup Product ◽

Lie Bracket ◽

Projective Resolutions ◽

Bar Resolution

We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH}^1(A)$ on $\operatorname{HH}^{\ast} (A)$.

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Hochschild cohomology, finiteness conditions and a generalization of d-Koszul algebras

Journal of Algebra and Its Applications ◽

10.1142/s021949882250147x ◽

2021 ◽

pp. 2250147

Author(s):

Ruaa Jawad ◽

Nicole Snashall

Keyword(s):

Hochschild Cohomology ◽

Finiteness Condition ◽

Koszul Algebras ◽

Finite Dimensional Algebra ◽

Koszul Algebra ◽

Dimensional Algebra ◽

Finiteness Conditions ◽

Finite Dimensional ◽

Projective Resolutions

Given a finite-dimensional algebra [Formula: see text] and [Formula: see text], we construct a new algebra [Formula: see text], called the stretched algebra, and relate the homological properties of [Formula: see text] and [Formula: see text]. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that [Formula: see text] has (Fg) if and only if [Formula: see text] has (Fg). We also consider projective resolutions and apply our results in the case where [Formula: see text] is a [Formula: see text]-Koszul algebra for some [Formula: see text].

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